ERRATA:
SHIMURA CURVES OF GENUS AT MOST TWO
JOHN VOIGHT
This note gives errata for the article Shimura curves of genus at most two [3].
These mistakes do not affect the main result; the list of curves is still complete and all curves have the correct signature (these signatures were independently verified, as in§5).
Errata
(1) Section 2, second paragraph: this is not a bijection, but a two-to-one map—
an embedding and its conjugate give the sameembedded finite subgroup.
(2) Lemma 2.1: The factor 2 from the previous mistake cancels the 2 in the denominator, so the corrected formula is:
eq = 1 h(F)
X
R⊂Kq
w(R)=2q
h(R) Q(R)
Y
p
m(Rp,Op).
(The formula that was implemented included both of these mistakes, the effect of which cancelled out.)
(3) Lemma 2.1: should beQ(R) = [NKq/F(R∗) :Z∗2F].
(4) Corollary 2.4: Same as the previous error: we have instead
eq = 1 h(F)
Y
p|D
1−
Kq p
X
R⊂Kq
w(R)=2q
h(R) Q(R).
(5) Lemma 2.5 is incorrect as stated. The detailed correction is given in the next section.
Embedding numbers
Lemma 2.5 is incorrect as stated. This mistake does not affect any other result in the paper; the list of curves is still complete and all curves have the correct signature (these signatures were independently verified, as in§5).
We give a complete and corrected statement and proof below. We retain the notation from§2. In particular, letRp=ZF,p[γp] and letπbe a uniformizer atp, and letfp(x) =x2−tpx+npdenote the minimal polynomial ofγp. Letdp=t2p−4np
and letk(p) denote the residue class field ofp.
We will use the following proposition in the proof; see Hijikata [1, §2] and Vign´eras [2,§III.3].
Proposition 2.3 (Hijikata [1, Theorem 2.3]).
Date: October 24, 2019.
1
2 JOHN VOIGHT
(c) Supposep|N. Let e= ordp(N)and fors≥e let E(s) ={x∈ZF/ps:fp(x)≡0 (mod ps)}. If ordp(dp) = 0then
m(Rp,Op) = #E(e).
Otherwise,
m(Rp,Op) = #E(e) + # img (E(e+ 1)→ZF/pe). The corrected lemma is then as follows.
Lemma 2.5. Let p be an odd prime. Suppose e= ordp(N)≥1, letr= ordp(dp), and letκ= #k(p).
• If r= 0, then
m(Rp,Op) = 1 + Kq
p
.
• If e < r, then
m(Rp,Op) =
(2κ(e−1)/2, ife is odd;
κe/2−1(κ+ 1), ife is even.
• If e=r, then m(Rp,Op) =
κ(r−1)/2, if ris odd;
κr/2+κr/2−1
1 + Kq
p
, if ris even.
• If e > r >0, then m(Rp,Op) =
0, if ris odd;
κr/2−1(κ+ 1)
1 + Kq
p
, if ris even.
Proof. Sincep is odd, without loss of generality we may assume that trd(γp) = 0, and henceE(s) is in bijection with
E(s) =
x∈ZF/ps:x2≡dp (mod ps) .
First supposer= 0. By Proposition 2.3(c), we havem=m(Rp,Op) = #E(e), and by Hensel’s lemma we see that #E(e) = 0 or 2 according asdp is a square or not inZF,p. In all other cases, we have the second case of Proposition 2.3(c).
Now suppose that e < r. The solutions to the equation x2 ≡ 0 (modps) are those with x ≡ 0 (modpds/2e). Thus #E(e) = κe−de/2e = κbe/2c and we see that # img (E(e+ 1)→R/pe) = κe−d(e+1)/2e, so m = 2κ(e−1)/2 if e is odd and m=κe/2+κe/2−1=κe/2−1(κ+ 1) ifeis even.
If e = r, then again #E(e) = κbe/2c. Now to count the second contributing set, we must solve x2 ≡ dp (modpe+1). If e = r is odd then this congruence has no solution. If instead e is even then we must solvey2 = (x/πr/2)2 ≡dp/πr (modp) where π is a uniformizer at p. This latter congruence has zero or two solutions according as dp is a square, and given such a solution y we have the solutionsx≡y (modπr/2+1) to the original congruence, and hence there are 0 or 2κr−(r/2+1)= 2κr/2−1solutions, as claimed.
Finally, suppose e > r > 0. If r is odd, there are no solutions to x2 ≡ dp (modpe). If r is even, there are no solutions if dp is not a square and otherwise
ERRATA: SHIMURA CURVES OF GENUS AT MOST TWO 3
the solutions are x ≡ y (modpe−r/2) as above so they total 2κr/2+ 2κr/2−1 =
2κr/2−1(κ+ 1).
The author would like to thank Virgile Ducet and to Steve Donnelly for bringing this error to his attention.
References
[1] Hiroaki Hijikata,Explicit formula of the traces of Hecke operators forΓ0(N), J. Math. Soc.
Japan26(1974), no. 1, 56–82.
[2] Marie-France Vign´eras,Arithm´etique des alg`ebres de quaternions, Lecture Notes in Math., vol. 800, Springer, Berlin, 1980.
[3] John Voight, Shimura curves of genus at most two,Math. Comp.78(2009), 1155–1172.
